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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 11th 2016
• (edited Jan 11th 2016)

At the old entry cohomotopy used to be a section on how it may be thought of as a special case of non-abelian cohomology. While I (still) think this is an excellent point to highlight, re-reading this old paragraph now made me feel that it was rather clumsily expressed. Therefore I have rewritten (and shortened) it, now the third paragraph of the Idea-section.

(We had had long discussion about this entry back in the days, but it must have been before we switched to nForum discussion, because on the nForum there seems to be no trace of it.)

• CommentRowNumber2.
• CommentAuthorDavid_Corfield
• CommentTimeJan 11th 2016

Back over here and here.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeMar 15th 2018
• (edited Mar 15th 2018)

briefly recorded some facts (here) on cohomotopy of 4-manifolds, from Kirby-Melvin-Teichner 12

• CommentRowNumber4.
• CommentAuthorUrs
• CommentTimeMay 18th 2018

added a few words in the Properties-section on the isomorphism between cohomotopy classes of smooth manifolds and the (normally framed) cobordism group in complementary dimension: here

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeOct 28th 2018

copied to here the remark about configuration spaces of points with labels in $S^n$ computing (twisted, unstable) cohomotopy (here)

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeMay 6th 2019

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019
• (edited Aug 23rd 2019)

added graphics (here) illustrating the unstable Pontrjagin-Thom isomorphism

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019

added also a graphics (here) illustrating the example of $\pi^n\big( (\mathbb{R}^n)^{cpt}\big) \simeq \mathbb{Z}$ under the PT-iso

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019

added also graphics (here) illustrating the $\mathbb{Z}_2$-equivariant version of the previous example.

Am adding the same illustration also to the respective discussion at equivariant Hopf degree theorem

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019
• (edited Aug 23rd 2019)

[duplicate announcement deleted]

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeAug 23rd 2019

further in this sequence of examples: added graphics (here) illustrating the equivariant Cohomotopy of toroidal orientifolds

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeAug 31st 2019

added one more graphics (here), meant to illustrated how the normal framing of the submanifolds encodes the sign of the “Cohomotopy charge” which these carry, under PT

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeSep 27th 2019
• (edited Sep 27th 2019)

• H. Sati, U. Schreiber:

where those graphics are taken from

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeOct 6th 2019

added these references on Cohomotopy cocycle spaces:

• Vagn Lundsgaard Hansen, The homotopy problem for the components in the space of maps on the $n$-sphere, Quart. J. Math. Oxford Ser. (3) 25 (1974), 313-321 (DOI:10.1093/qmath/25.1.313)

• Vagn Lundsgaard Hansen, On Spaces of Maps of $n$-Manifolds Into the $n$-Sphere, Transactions of the American Mathematical Society Vol. 265, No. 1 (May, 1981), pp. 273-281 (jstor:1998494)

• CommentRowNumber15.
• CommentAuthorUrs
• CommentTimeOct 20th 2019
• (edited Oct 20th 2019)

prodded by Dmitri (here) I have added a remark on terminology (here). In the course of this I ended up considerably expanding the Idea-section; now it has also a subsection “As the absolute cohomology theory” (here)

• CommentRowNumber16.
• CommentAuthorDmitri Pavlov
• CommentTimeOct 20th 2019
Thanks! In the table of flavors of Cohomotopy, shouldn't we also have differential Cohomotopy?
• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeOct 20th 2019

Yes! Eventually we need a higher dimensional table. Or a table of tables.

Personally, of course I am eager to go full blown into twisted equivariant differential Cohomotopy of super orbifolds. And Vincent has a bunch of ideas for what to do. But to make sure not to be barking up the wrong tree, we’d first like to finish one or two further consistency checks in the “topological sector” first.

But that’s just me. If you want to go ahead creating more nLab entries on further variants, please do.

• CommentRowNumber18.
• CommentAuthorDavid_Corfield
• CommentTimeOct 23rd 2019

• Peter Franek, Marek Krčál, Cohomotopy groups capture robust Properties of Zero Sets via Homotopy Theory, (slides)
• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeOct 23rd 2019

Thanks for the pointer. That made me add also the article that it’s based on:

• Martin Čadek, Marek Krčál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, Uli Wagner, Computing all maps into a sphere, Journal of the ACM, Volume 61 Issue 3, May 2014 Article No. 1 (arxiv:1105.6257)
• CommentRowNumber20.
• CommentAuthorDavid_Corfield
• CommentTimeOct 24th 2019

I think the talk is closer to

so have added that. It seems to rely on values of a function bounded away from $0$ being mapped to a sphere.

• CommentRowNumber21.
• CommentAuthorUrs
• CommentTimeOct 28th 2019

Thanks! Interesting. I am adding cross-links with persistent homology (in lack of a general mathematical notion of “persistency” of which these two are examples(?))